space-time geometry of a torus for a lorentz- and conformal

63

African Journal of Physics Vol.3, 2010.

SPACE-TIME GEOMETRY OF A TORUS FOR A LORENTZ- AND CONFORMAL-INVARIANT STRING THEORY WITHOUT

DIVERGENCES†

A.O.E. Animalu and N.C. Animalu

Dept. of Physics and Astronomy, University of Nigeria, Nsukka, Nigeria &

Int. Centre for Basic Research, Limpopo St. FHA, Maitama, Abuja, Nigeria

email: [email protected]

Abstract

From an analysis based on linguistic/projective geometry of the meeting point(s)

of the quantum culture(s) of the East and West and a pre-historic African(Igbo-

Ukwu) culture that built a hyperdoughnut, i.e. a torus in bronze cast (c.10,000-

5,000 BC), we present a strong case for ab intio use of the space-time geometry

of a torus for the construction of Lorentz- and conformal-invariant string theory of

everything without divergences. From this perspective, we show firstly, that the

Nambu-Goto action integral associated with an element of surface area of the

torus generates an Eulerian beta-function of the type envisaged in the Veneziano

(dual resonance) model of hadrons, and that the conformal (non-unitary

topological) mapping of the torus onto a two-sheeted (Riemann) plane provides

new insight into the relation of modular functions to the theory of interacting

strings. Secondly, we develop a non-phenomenological purely geometric

approach to compactification of M-theory in 11-dimensions and superstring

theories in 10- and 26-dimensions based on the principle of duality in projective

space-time geometry of the torus which is equivalent to supersymmetry

transformations between vectors and antisymmetric tensors in 4-dimensional

Minkwoski space. Compatibility of particle-antiparticle (boson).open-string

variables (with internal SU(2) symmetry) and three-fermion close-string variables

equivalent to plane coordinates (with internal SU(3) symmetry) then leads to

Maxwell’s type of consistency equations and current conservation as well as a

possible explanation of the vanishing of the cosmological constant. Thirdly, we

express the metric tensor of the 5-dimensional superspace of a torus in terms of

†African Journal of Physics Vol. 3, pp.63-93, (2010)

ISSN: PRINT: 1948-0229 CD ROM:1948-0245 ONLINE: 1948-0237

64


anticommuting Dirac matrices and relate it to the 32x32-matrix representation of

creation and annihilation operators of the group SO(2N) for N=5. Finally we

outline how the present theory can be tested experimentally and draw conclusions.

1. INTRODUCTION

One of the outstanding problems of contemporary quantum physics culture is

to establish a mathematically self-consistent relation between the usual

mind/body problem of consciousness in the universe and the wave/particle

duality problem of quantum mechanics, i.e. Heisenberg’s uncertainty principle,

or in other words, the problem of unifying the descriptions of the whole universe

(macrocosm) and its minutest parts (microcosm). The famous Schrodinger cat

experiment in which the cat always responds according to the expectation of the

observer has led to the conclusion that the cat is conscious and aware of the

mind-set of the observer, and responds in consonance with that mindset as if it has

its own “mind” that resides in the “wave field” associated with the motion of its

parts. As recalled by Pauli (1958a, p.260) in his Writings on Physics and

Philosophy, “after Kepler(1600) had shown that the optical images on the retina

are inverted in relation to the original objects, he baffled the scientific world for a

while by asking why then people did not see objects upside down instead of

upright”. No wonder Alva and Heidi Toffler (1981) on p.46 of their book entitled

The Third Wave made the following remarkable statement: “Every civilization has

a hidden code – a set of rules and principles that run through all its activities like a

repeated design”. The significance of this statement emerges when we classify the

revolutionary epochs of modern physics into three distinct waves.

The First Wave Physics began in Renaissance Europe with the publication of

Kepler’s three empirical laws of planetary motion in the year 1600, and Newton’s

synthesis of these laws in the 1660s into the deterministic laws of classical

mechanics that ushered in the Machine Age. The “hidden code” that led to the

synthesis lay in the Judeo-Christian Linear creation axiom which found its way

into Renaissance thought through “the Cartesian rectilinear grid …[which] is

suitable for analysis of the world into separately existent parts (e.g. particles and

fields)” as stated by D. Bohm(1980) on p.xiv of his book entitled Wholeness and

the Implicate Order. That this hidden code ran through Newton’s synthesis is

evident in the questions he posed at the end of his researches in Optics, following

his apparently futile attempt to construct a linear and mechanical theory of light.

The atrophy of the First Wave Physics, which some may regard as its triumph,

began with the parting of the ways between science and religion in the wake of

the trial of Galileo (c. 1613) by the Roman Church which made scientists,

thereafter, to shun the twilight zone between science and religion. Yet, scientists

65


continue, consciously or unconsciously, to roam in the twilight zone in search of

an understanding of man himself.

The Second Wave Physics began three hundred years later with the birth of

quantum mechanics of atoms and molecules in 1900. Quantum mechanics still

retained the rectilinear grid (extended slightly in Einstein’s theory of relativity to

the curvilinear grid) in that (see, p.66 of D Bohm(1980): i) The fundamental laws

of the quantum theory are to be expressed with the aid of a wave function which

satisfies a linear equation (so that solutions can be superposed linearly), and ii)

All physical results are to be calculated with the aid of certain ‘observables’

represented by Hermitian operators, which operate linearly on the wavefunction.

But Heisenberg’s indeterminacy principle shattered the determinism of

Newtonian Mechanics, while relativistic quantum mechanics revealed the

inconsistency of the atomistic nature of the linear Cartesian order with the

interconnectedness or wholistic order of the known families of the elementary

particles, as described by Fritjof Capra(1978) in his book entitled The Tao of

Physics. The achievement of the Second Wave Physics lies in the triumphs of

electronics technology that has ushered in the Computer Age – a triumph of

silicon over steel in the creation of the new information wealth of the developed

nations. The atrophy of second Wave Physics began with the parting of ways

between the atomistic and the wholistic order in the wake of the breakdown in

communication/dialogue between the eminent founding fathers of the quantum

theory, Bohr and Einstein, summed up in the justly famous Einstein dictum: “God

does not play dice with the universe”.

The Third Wave Physics is currently under way. Its objective is to achieve

a synthesis of the Cartesian atomistic order and the interconnectedness or

wholistic order found in living things or the so-called self-organizing or growing

systems; or as Nick Herbert (1985) puts it on p. 249 of a book entitled, Quantum

Reality, Beyond the New Physics: “One of the greatest scientific achievements

imaginable would be the discovery of an explicit relationship between the

waveform alphabets of quantum theory and certain human states of

consciousness” Since consciousness is both wholistic and atomistic, an even

greater achievement may lie, not only in the quantization of human states of

consciousness but in the geometrization of quantum theory in order to make it

comprehensible not only to the human senses but also to the mind. For according

to Plato: “what is perceptible to the senses is the reflection of what is intelligible

to the mind”. In other words, instead of the Judeo-Christian order expressed by

the Pauline dictum (Romans 1:20): “Per visibila ad invisibila”, which has often

been interpreted in the Western World to mean that the invisible or spiritual order

is subservient to the material order, Third Wave Physics allows for the dual

principle: Per invisibila ad visibila, i.e. a material order subservient to the

66


spiritual order. It is no wonder then that Western philosophers and physicist have

long been wondering how the spiritual world, so obviously real, could so

completely elude detection by the increasingly sensitive instruments of science.

The answer may well lie in the inadvertent obliteration of the wholistic view –

like the proverbial jealous dog carrying a piece of bone in the mouth and arriving

at the bank of a river. On seeing its own distorted image with bone in its mouth

reflected from the river (and lacking the human consciousness of image

recognition), jealously opens its mouth to bark/grab the bone from the other dog

reflected in the water, and lost the bone into the depth of the water, in apparent

confusion of greed and “chasing its own shadow”. Put in other words, a physicist

solving Einstein’s equations of general relativity soon arrives at a singularity or a

“boundary” of the universe. On looking at the distortion of his field equations by

the “image “ of his universe in the boundary, the physicist abandons his

deterministic equations by jumping into the sea of indeterminate “quantum

fluctuations”, where he loses consciousness of the whole and talks only about the

“early universe”. To regain consciousness, the Third Wave Physicist has to

geometrize the quantum fluctuations by eliminating the boundary through

topological folding of his Cartesian cubic grid into a torus.

It is this paradigm of Third Wave Physics espoused in Kaku’s(1995) ten-

dimensional hyperspace view of the (superstring) theory of everything (TOE)

(unifying the four basic (electromagnetic, weak, strong and gravitational) forces

in nature and Steven Hawking’s(1986) torus view of the universe (based on his

use of quantum fluctuations to eliminate the black-hole singularity of Einstein’s

general relativity theory of gravitation) that we wish to recast from the African

perspective described in the preceding paper by Animalu and Acholonu(2010).

Our interest (see, Fig. 1 below) is to determine how the facial scarification

(“ichi”) linguistic geometry of the alphabet system of the spoken (Igbo) language

of the prehistoric African (Igbo-Ukwu) culture that produced an “enigmatic tyre”

(i.e. a torus) in the bronze cast escavated in the 1950s by the British archaeologist,

Thurston Shaw(1950) provides meeting point(s) of various (East, West and

African) cultures with modern physics especially the string theory of

supergravity, called the M-theory, and its further development into the superstring

theory of everything (see, especially, the series of articles in Cambridge

University Press published Superstrings, A Theory of Everything? edited by

Davies and Brown(1988); Michio Kaku’s (1995) book entitled Hyperspace with

subtitle A Scientific Odyssey Through Parallel Universes, Time Warps, and the

10th

Dimension; and Time Magazine Dec. 31, 1999 edition on Albert Einstein as

Person of the Century). While the M-theory is constructed in 11-dimensional

hyperspace geometry inhabited by weird objects called branes, superstring theory

distinguishes an open string model with gauge group SO(32) and a closed

(“heterotic”) string model with gauge group E8xE8 for grand unification of

67


strong and electroweak forces with gravity in hyperspaces of 10- and 26-

dimensions which will be elucidated by linguistic geometry.

Apart from the fact that current theories of everything have not accounted

for the quantization of Kaku’s hyperspace/torus in such a depth as to explain the

intriguing hexagonal grid pattern on the African (Igbo-Ukwu) bronze cast torus,

Fig.1: (i) Niels Bohr’s coat-of-arms (Rozental 1967); (ii) Chinese I-Ching (cf

Gell-Mann & Ne’eman(1963) 8x8 representation of SU(3)); (iii) Torus buit

from equal matter and antimatter according to “big bang” theory of creation

& string theory of everything(TOE); (iv) Facial (“ichi”) scarification/linear

writing in African (Igbo-Ukwu) bronze face of the Sun (Acholonu, Plate 41,

TLBA 2009), (v) Plato’s model of the earth (c.300 BC);(vi) Robert

Fludd’s(1617) medieval symbol of man as cosmic centre; (vii) Kepler (1600)

cosmic harmony; (viii) Egyptian pyramid & its two-dimensional (pentagon)

projection on a plane; (ix) Step cylinderical pyramid in Nsude, Nigeria & (x)

Igbo-Ukwu bronze torus with distinctive cube-hexagon grid (TLBA p.447).

68


relativistic quantum field theory is further plagued by the existence of two

singularities in Einstein’s special relativity theory, namely the singularities in

the Lorentz transformation at cv and in Einstein’s addition theorem at 2cvV for group velocity (v) and phase velocity (V) associated with the

wave/particle duality problem. Taken together, these singularities imply the

existence of faster-than-light particle states with velocities, c/vcV 2 for

cv and violation of causality which is forbidden. To deal with these problems,

we are compelled to found Einstein’s principle of relativity ab initio on the space-

time geometry of a torus as we proceed to present in this paper outlined as

follows.

A reformulation of Einstein’s principle of relativity on a torus will be

presented in Sec. 2.2 : it will have the consequence that it leads not only to a

realization of the beta-function discovered “accidentally” by Veneziano(1974),

but also to a theory of conformal (non-unitary) transformation of the torus onto a

two-sheeted (Riemann) plane with a number of interesting physical features.

Among the physical features of the conformal (non-unitary) transformation is a

projective geometric method for compactifying hyperspace manifold down to the

usual 4-dimensional manifold of point-particle field theories. This will be

presented in Sec. 3 along with an extension of the method provided by geometric

principle of duality to compactification of both 10- and 26-dimensional

manifolds. In Sec. 4 we shall develop extended relativity on the torus and

determine the constraints on the algebraic dimensions of a second-quantized

theory from the 2Nx2

N(=32x32)-matrix representation of creation and annihilation

operators that generate the group SO(2N) for N=5. In Sec. 5 we shall outline how

the present theory could be tested and draw conclusions.

2. FORMULATION OF STRING THEORY ON A TORUS

2.1 Reformulation of Einstein’s Principle of Relativity on a Torus

In order to enable the (Cartesian) cubic frame of reference to be used in

characterizing quantization of space-time geometry so as to account for the

distinctive hexagonal grid on the surface of the Igbo-Ukwu bronze torus(see, Fig.

2) in ab initio formulation of Einstein’s principle of relativity, we presume that

such a torus (built from equal parts of matter and antimatter, in accordance with

the “big bang” theory) is defined in 3-dimensional (x,y,z)-space by the parametric

equations (Spinger 1957, p. 34):

,sin,sin)cos(,cos)cos( ctzctsyctsx (2.1.0)

69


where t is the time, c the speed of light, s the proper distance, the meridian

angle, and the latitude angle.

On eliminating and from (2.1.1a) we obtain an equation of the torus in the

forms.

sin2222222 ctszyxtcs (2.1.1a)

i.e., .2 222222222 ztcszyxtcs (2.1.1b)

or, .4)( 22222222222 ztcszyxtcs (2.1.1c)

or, .0)()(2 2222222222224 zyxtczyxtcss (2.1.1d)

Fig.2 : (i) Igbo-Ukwu Bronze Torus View of a universe built from

equal parts of matter and antimatter in (ii) complementary

(cube & hexagon) frames of reference defining (iii) a cube-

hexagon-pyramid/torus hyperspace.

70


A generalization to ellipsoidal torus having generalized Lorentz symmetries may

be incorporated via the substitutions, ,1xbx ,2 yby , 3zbz ctbct 0 ,

in the above equations. In the canonically conjugate energy-momentum

0,1,2,3)( p and mass variables, the analogous equations are:

,sin,sin)cos(,cos)cos( 030201 pppmcppmcp (2.1.2a)

i.e. .2 2

3

2

0

2

3

2

2

2

1

2

0

22 ppmcppppcm (2.1.2b)

or .)(4)( 2

3

2

0

222

3

2

2

2

1

2

0

22 ppmcppppcm (2.1.2c)

or, .0)()()(2)( 22

3

2

2

2

1

2

0

2

3

2

2

2

1

2

0

24 ppppppppmcmc (2.1.2d)

We now observe that Eq.(2.1.1b) is invariant under the usual Lorentz

transformation:

),v(' ,' ,' ),)/v((' tzzyyxxzcctct (2.1.4)

where 22 /v1/1 c . That is, as can be verified directly,

222222222 ''4'''' s ztcszyxtc .

Moreover, if in Eq.(2.1.1d) 0s but

022222 zyxtcxx with )1,1,1,1.()( Diag (2.1.5a)

we have the “extended” relativity relation

xxgzyxtcs )( 222222

with )1,1,1,1.( Diagg (2.1.5b)

and ),,,(),,,( 3210 zyxctxxxx . This guarantees that Einstein’s relativity

principle founded on the Lorentz invariance of Eq.(2.1b) can be used as an

heuristic aid in the search for general laws of nature whose underlying geometry

is the torus defined by Eqs.(2.1.0).

We also observe, by using the Dirac method of removing the square root

from Eq.(2.1.2b), that one can write down a 4-component spinor wave equation

.)(2)( 3300

2

3

2

2

2

1

2

0

22 ppmcppppcm (2.1.6)

where g2 . This is a 4-component O(4,2) wave equation which

determines a more general conserved current including the so-called “convective”

71


currents proportional to the total momentum of the particle-antiparticle system

(Animalu, 1971) than the one determined by the conventional Dirac equation.

From Eq.(2.1.1a) the quantization of the torus into a lattice results from

n

ctscts

if , 0

,)(n if ),(2 sin)(2 2

1

(2.1.7a)

i.e.,

nzyxsct

zyxsct

if , 0 & 0)(

)(n if ,0)(22222

212222

(2.1.7b)

where n = 0, 1, 2,… Since the only values of the angle compatibles with

perfect translational symmetry of a crystal lattice in three-dimensional space are

those for which integer )1cos2( , we have ,2 n where 1,2,3,4,6n

include cubic and hexagonal lattices, but not pentagonal lattices (Ziman,1960,

Animalu 1977). The visual images of the types of geometric objects represented

by Eqs.(2.1.7) can be constructed by rewriting the first set of equations as follows

,0)()( 22222222 sctzyxsctzyx (2.1.8a)

so that, for )(n21 , a pair of concentric spheres in r

-space of radii sct

define a spherical shell of thickness, s2 , with one sphere circumscribing the cube

and the other sphere circumscribing the hexagon as shown in,Fig.3: This may be

termed the “black hole” solution associated with “Big Bang”.

Fig.3: (i) two concentric spheres of radii )( sct

defined

Eq. (2.1.8a); (ii) (u,v) generators of the “string” solution

(2.1.8b) analogous to (iii) Kaku’s communication “worm

hole” ; (iv) –(vi) Symmetric objects (tetrahedron, square &

hexagonal pyramids) associated with the quantized

values of 2 n for n=3,4,6.

72


.The second of the two equations in (2.1.7b) is a ruled quadric surface with real

)vu,( line-generators in 3-dimensional projective space with homogeneous

coordinates ( ),,, yxsct given by

n

yctxs

xsyct

xctys

ysxct

if

0v

v

0u

u

2

1

2

1

(2.1.8b)

which may be termed “string” or “worm hole” solution as shown in Fig.3(ii) &

(iii).

Consequently, the geometry of the torus is a promising one for

formulating not only “extended” relativity principle and quantization of space-

time but also the Nambu-Goto action principle in string theory to which we now

turn.

2.2 Nambu-Goto Action Principle for a Torus

We construct the metric (E,F,G) for the 2-dimensional ( , )-space on the

torus by noting that, by virtue of Eqs.(2.1.1), we have

d

d

zz

yy

xx

dz

dy

dx

(2.2.1a)

i.e., 22222 2 GddFdEddzdydx (2.2.1b)

.)cos ()()()(

,0)()()()()()(

,)()()(

2222

22222

ctszyxG

zzyyxxF

tczyxE

Thus, the action defined by Nambu(1970) and Goto(1971) representing the area

of the curved surface in ( , )-space is given by the surface integral,

ctctsddFEGddA )cos (2 . (2.2.2)

73


It is apparent from this that minimizing of the action (A) is tantamount to

minimizing both the time (t) and the proper distance (s), as expected in Einstein’s

general relativity equations that determine the (geodesic) path of particles in

space-time. But we have, in addition, a non-trivial dependence on the latitude

angle ( ), which can be viewed from the following two perspectives.

2.3 Beta-Function and Conformal-Invariance

Firstly, if the limits of the -integration are from 0 to 2/ , then

Eq.(2.2.2) generates an Eulerian beta-function defined by

2/

0

2/

0

1212 cos2 sin cos2),(

ddpqB qp, (2.3.1)

where 1p and 21q . The interest of this lies in the fact that it suggests an

origin of the beta-function discovered by Veneziano(1974), which has been

described as “an answer looking for a question”.

Secondly, a theory of conformal mapping (i.e. a non-unitary

transformation) of the torus onto a two-sheeted (Riemann) plane arises by

expressing Eq.(2.2.1b) in the form:

212

1

)(cos 2 2222 ddctsGddFdEd (2.3.2)

as prescribed Springer (1957, p.34), where and is given by

21

21

0tan

tanlog)/()cos /(

Acts

ActsActctsctd (2.3.3)

(Beyer (1987, p. 266), where 21

222 stcA . This means that ),( are the

isothermal coordinates (Springer 1957, p.19) on the torus which maps the torus

(as and vary between and into a rectangular region (

and ActAct // ) in the ),( plane. Note that space-like regions

)0( 2 s require a complex-valued .

On the light cone )0( s , Eq.(2.3.3) reduces to 21

21 tantanh which

must reduce to the result obtained by computing Eq.(2.3.3) in the form

zct

zctztcctdzd

z

log)/( cos / 21222

00

(2.3.4)

74


i.e., sin/tanh ctz (2.3.5)

by virtue of Eq.(2.1.0). Thus the conformal transformation may be written

ie

zct

zctiW 2

21 log or )'sin(tanh iW (2.3.6)

where ' is appropriately defined, so that Eq.(2.3.6) is an analytical continuation

of Eq.(2.3.5) into the complex plane. Then we may re-parameterize the torus

defined by Eq.(2.1.1b) in the form

sinh,sinh,sinh,cosh),,,( iszyxct (2.3.7)

which agrees with Eq.(2.3.5) and determines from the cross-ratio

2

))((

))((ie

yxzct

zxyct

,

i.e. .))((

))((log 22(

21

ie

yxzct

zxyctiW (2.3,8)

The interest of the conformal mapping is that it leads to a number of physical

characteristics of string theory. Firstly, it provides a realization of the invariance

of the string action under the re-labelling of the space and time coordinates on the

world sheet having the topology of the torus defined by ( , ) of Eq.(2.1.1a) and

by ( s , ) in Eq.(2.3.7). Accordingly, Eq.(2.3.2) becomes, in terms of ( s , ):

2FEGdsdA (2.3.9)

where the (E,F,G) are given by the expressions (with )),,( zyxr

22

2

,0

,1

srrtcG

rsrtcstcF

srsrstcE

(2.3.10)

Thus, we obtain the following result,

2

21 ssdsdA , i.e. 2/2 sA (2.3.11)

which is an expression of the isothermal coordinate ( ) in terms of s and the area

(A) [cf, Goursat (1959, p. 218)]. By comparing Eqs.(2.3.8) and (2.3.11), we infer

from the periodic (many-valued) property of the logarithmic function that when

75


changes by )1(2 N , with N=0,1,2,…, being an integer, the corresponding

change in A is given by

),1(2/2 2 NsA i.e., )1)(/(2 NGm (2.3.12)

where AG is the Newtonian constant of gravitation and 22 /1 sm

characterizes a mass (m), in units such that 1 c . This is the mass spectrum

of the first quantized string theory(Gross, et al (1985), Candelas and

Horwitz(1985)) for a bosonic string in 26d space-time dimensions. For N=0,

Eq.(2.3.12) gives )/(2 Gm , which represents a tachyon and for N=1, it gives

02 m which is a massless state usually associated with the spin-2 graviton.

2.4 Interacting Open and Closed Strings

Another interesting physical feature of the conformal mapping of the torus

onto the W-plane defined by Eq.(2.3.6) arises from the so-called level curves of a

function of a complex variable, )(Wf defined (Hardy 1958, p. 484) as curves for

which )(Wf , called modular function, is constant. The level curves on the torus

arising from cW /vtanh in the W-plane are defined, from Eq.(2.3.6), by

./v)'sin( Kci (2.4.1)

These curves (see, Fig. 4) may be interpreted as a motion along the z-axis of the

torus in which a closed string expands (as K increases) and breaks up into two

open strings.

Fig. 4: Level curves of Eq.(2.4.1) (after Hardy(1958, p. 484,

Fig. 56). The curves marked I-VIII correspond to K=0.35, 0.50,

0.71, 1.00, 1.41, 2.00, 2.83, 4.00

76


Evidently, we may look upon Eq.(2.4.1) as the result of an analytical

continuation of the Euclidean beta-function defined by Eq.(2.3.1) (cf.

Phillips(1972, p.111 Eq.(15) ):

).'sin(2 cos2)';,(

'

0

idiqpB

i

(2.4.2)

If the lower limit of integration is cut off at a non-zero value of 0 say, then

Eq.(2.4.2) may be replaced by

]sin)'[sin(2),';,( 00 iiqpB (2.4.3)

This leads to the level curves of Ci )'sin( , ( 0sinC being a positive

constant), which are shown for 1C in Fig. 5, and may be interpreted as motions

in which two closed strings on the torus join (as K increases) to form a new

Fig. 5: Level curves of

Ci )'sin( : for C=0.5 (after

of Hardy(1958, p. 484, Fig. 58).

The curves marked I-VII

correspond to K=0.29, 0.37,

0.50, 0.87, 1.50, 2.60, 4.50, 7.79.

Fig. 6: Level curves of

Ci )'sin( for C=2 (after

Hardy(1958, p. 484, Fig. 59).

The curves marked I-VII

correspond to K=0.58, 1.00,

1.73, 3.00, 5.20, 9.00, 15.59.

77


(larger) string and subsequently break up into two open strings. A similar case

corresponding to 1C is shown in Fig. 6. If 1C the curves are the same as

those of Fig. 4, except that the origin and scale are different.

2.5. Internal SU(2) Symmetry

An alternative way of looking at the conformal mapping of the torus onto

the W-plane in the general ( 0s ) case defined by Eq.(2.3.8) is in terms of an

internal SU(2) symmetry embodied in the invariance of the cross-ratio of the

space-time coordinates. On putting ),,,(),,,( 3210 xxxxzyxct and

constant2 We , Eq.(2.16) may be rewritten in the form (with 4/ ):

0))(())(( 2130

2

3120 xxxxexxxx W (2.5.1a)

i.e., 0)()())(1( 12301320

2

3210

2 xxxxxxxxexxxxe WW

or, ,0)()())(1( 2

12

2

03

2

32

2

02

22

23

2

01

2 ffffeffe WW (2.5.1b)

where xxf . Thus the f’s behave like the components of an

antisymmetric tensor field,

exexf with ),1,1,1,1(),,,( 3210 eeee (2.5.2)

which is invariant under the SU(2) gauge transformation defined by

)1(,',' 1221221122211211 aaaaeaxaeeaxax (2.5.3)

This is equivalent to the usual linear fractional transformation,

)/()(' 22211211 axaaxaxx (2.5.4)

which leaves the cross-ratio ))(/())(( 21303120 xxxxxxxx invariant. The

f’s also obey the orthogonality (or transversality) law,

,0120331022301 ffffff (2.5.6)

which shows that f can be identified with the free electromagnetic field, or

more generally, a Yang-Mills field.

Two important features of the invariance of the cross-ratio of the

coordinates under the linear fractional transformation (2.5.4) are worth noting.

Firstly, there are 24 possible cross-ratios (Semple and Kneebone 1952, p.47) of

78


the pair ( xx , ) with respect to the pair ( xx , ) of coordinates defined as

follows:

(2.5.5) ;)(

)(/

)(

)(,;,

02

01

32

310321

xx

xx

xx

xxxxxx

But there are only 6 distinct cross-ratios, namely

0321 ,;, xxxx , 1,;, 0231 xxxx ,

1,;, 0312 xxxx , (2.5.6)

11,;, 0132 xxxx ,

1

1,;, 0213 xxxx ,

1,;, 0123

xxxx

The number 24 frequently pops up in connection with Ramanujan’s modular

functions in dual resonance model.

Secondly, suppose that )/2,,0,()',',','( 21

3210 pmGxxxx where G is

the Newtonian constant of gravitation and pm the proton mass (in units such that

1 c ). Then

21

/2)'')(''/()'')(''( 21303120

2 Gmxxxxxxxxe p

W

i.e., )/2log( 21

21 GmW p . (2.5.7)

This is satisfied when 3/12 W , with 137/1 being the electromagnetic

fine-structure constant, as may be verified by putting cmG 3321

106.1 and

cmmp

14101.2/1 to find

(2.5.8) 4.137)102.8log(3)/2log(3/1 1921

Gmp

It follows from this that the self-mass of the proton computed in the

renormalization theory of quantum electrodynamics with 21

/2 G as (Planck)

cut-off mass is finite:

MeVmGmmm ppp 1492/)/2log()2/3( 21

. (2.5.9)

This is of order MeVm 140 , which is the mass of the pion cloud surrounding

the physical (renormalized) proton. Such a finiteness is an important feature of

string theory.

79


3. GEOMETRIC APPROACH TO COMPACTIFICATION

3.1 Compactification of the D=10 String Theory

Let us now return to Eq.(2.3.8) involving a relationship between 3D

dimensions of the coordinates (x,y,z) and 2D dimensions of the coordinates

),( , in order to abstract a general method for compactifying an arbitrary D-

dimensional space with coordinates 1)-D0,1,...,( ),( yx to the d-

dimensional space with coordinates 1)-d0,1,...,( y :

, i.e., ,)/(

dydygdxdxdyyxdx (3.1.1)

where )/)(/(

yxyxg . For the important case of 10D with

coordinates 0,1,...,9)( ),( yx and 4d with coordinates

0,1,.2,3)( , xy , we may take ),,(),( 030201654 fffxxx and

),,(),( 123123987 fffxxx , where xxf as discussed above. A more

elaborate geometric approach to compactification based on geometric principle of

duality and representation of conserved currents as coordinates(Sommerville

1959) is required for compatification of D=26 string theory to which we now

turn.

3.2 Geometric Principle of Duality

The geometric principle of duality states that a light cone may be

generated in either of two ways: either as the locus of points 0,1,2,3)( , x in

4-dimensional Minkowski space ( 4M ) in which it is represented by the

homogeneous algebraic relation,

02

3

2

2

2

1

2

0 xxxxxxg

. (3.2.1a)

Or (dually) as the envelop of its complex of tangent lines X in 6-dimensional

tangent space ( 6B ) in which it is represented by the homogeneous algebraic

relation (Semple and Kneebone 1952, p. 313) or Sommerville (1959, p.374):.

0)( 2

12

2

31

2

23

2

03

2

02

2

01 XXXXXXXXgggg

(3.2.1b)

80


This principle is illustrated in Fig. 7 in which a circle is generated in either of two

ways: either by the motion of points or the turning of tangent lines.

The relationship between Eqs.(3.2.1a) and (3.2.1b) is tantamount to

supersymmetry (SUSY) between a vector x and an antisymmetric tensor

XX . The other SUSY transformation of interest below is between

bosons and fermions, (represented respectively as quark-antiquark )q(q 11 (open

string) and three-quark )qq(q 321 (closed-string) systems). The pair ( x , X )

defines a 10-vector space which undergoes by virtue of Eqs.(3.2.1a) and (3.2.1b)

spontaneous compactification to 64 BM .

To relate x and X mathematically, we consider the usual Euler’s

transformation of a position vector )x,x,(xr 321

which is equivalent to a

rotation through a definite angle about some definite line defined by the

direction of a unit vector )e,e,(ee 321

in 3-dimensional space. This

transformation is given by the formula (Sommerville 1959, p.39).

)cos-(1)r.()sinr(cosr'rr aaa

(3.2.2a)

or, in infinitesimal form (first order in small expansion),

,)r(r

a i.e., ,)X,X,(X),,( 123123321 xxx (3.2.2b)

Fig. 7: Dual generation of a circle either as

locus of points or envelop of tangent lines

81


where )(Xij ijji xaxa are components of an antisymmetric tensor of second

rank in 3-dimensional space. This shows that ix has an antisymmetric tensor

(SUSY) partner ijX with 3=(3/2)(3-1) independent components equal to the

number of generators of the Lie algebra of SU(3); and hence )X,(x iji form a 6-

vector space which admits a spontaneous compactification to 33M B , where 3M is the usual 3-dimensional space of points ( ix ) and 3B is its dual 3-

dimensional space of “surface tensor” ( ijX ) defined by (Pauli 1958b, pp.30-31).

This result has a number of additional features in 4-dimensional space (4M ) of points x and 6-dimensional space ( 6B ) of “surface tensor” X with

6=(4/2)(4-1) independent components equal to the number of generators of the

Lie algebra of the Poincare group, SO(4) or SO(1,3), such that )X,(x form a

10-vector space which admits a spontaneous compactification to 64M B .

Suppose that x is the position 4-vector of a point particle ( e say) in 4M tied by

a string to a point-antiparticle ( e say) located at ax , a being a unit 4-

vector, so that the ee boson (open string) system may be characterized by the

surface vector,

'' XxxxxX (3.2.3)

Then by virtue of the relation (Pauli 1958b, p. 30, Eq.(46))

0'''''''' 120331022301 XXXXXXXX (3.2.4)

obeyed by s surface tensor in 4M , we have on making the substitutions,

,'','',''

,'','',''

120312310231230123

120303310201230101

XXXXXXXXX

XXXXXXXXX

(3.2.5)

the SO(3,3)-invariant relation cited in Eq.(3.2.1b):

02

12

2

31

2

23

2

03

2

02

2

01 XXXXXXXXR

(3.2.6)

where, by using Eq.(3.2.1) as a definition of the metric tensor g ,

ggggR (3.2.7)

is a (constant) curvature tensor. This implies that, in a purely geometrical theory,

an ee (open-string) boson admits a 10-vector space representation of the form

82


),( Xx where x is the 4-vector position of 4M in and X is antisymmetric

tensor (SUSY) partner related to it by Eqs.(3.2.3) and (3.2.5) with the independent

components of X defining a 6-vector space ( 6B ). Moreover, the constant

curvature tensor defined by Eq.(3.2.7) is the “metric tensor” of the 6B of X :

this explains the vanishing of the cosmological constant of the 10-vector space

under compactification to 64 BM .

In like manner, a three-particle ( 321 qqq ) fermion system admits a (closed

string) representation of the form ),,,,,( ZzYyXx where zyx ,, are

the respective coordinates of 1q , 2q , 3q in 4M and

yxyxXxzxzXzyzyX , , (3.2.8)

are the “surface tensors” or Plucker’s line coordinates ((Pauli 1958b, p.31

footnote (60a)) of the strings joining the respective particles pairs,

211332 ,, qqqqqq , to form the closed (fermion) string in question. In order to

establish the connection of this representation with the 26-dimensional tangent

space envisaged in the “heterotic” closed string model(Nambu 1970,Goto 1971)

we also introduce the “volume tensor” ((Pauli 1958b, p. 30) or plane-coordinates

associated with the three points zyx ,, , via the determinant,

.

zyx

zyx

zyx

X (3.2.9)

This third rank antisymmetric tensor has only 4 independent components

),,,( 012301230123 XXXX which may be denoted by )3,2,1,0( , J termed the

plane-coordinates of the three points in question. Accordingly, the 321 qqq closed-

string system may be represented geometrically by the set of coordinates

),,,,( JZYXx forming a 26-vector defining a representative point of the

26-dimensional tangent space, which has spontaneous compactification of the

form, 46664 ~MBBBM where 4~

M is the space with plane coordinates (

J ).

3.3 Compatibility Relations and Current Conservation

The most important consequences of the above mathematical

compactification scheme arise from certain compatibility relations due to the

83


existence of SUSY transformations between 1q (fermion) and 11qq (boson) on

one hand, and between 11qq (boson) open-string and 321 qqq (fermion) closed-

string systems on the other hand. In the language of the geometric principle of

duality, the compatibility in question is that the line ( xxxxL )

representing the boson 11qq must meet the plane ( J ) representing the fermion

321 qqq at their common (intersection) point ( x ) representing the fermion 1q in

4M , which is the case provided that the coordinates zyxx , , , are linearly

dependent, and hence obey the determinantal condition:

33221100

3210

3210

3210

3210

0 JxJxJxJx

zzzz

yyyy

xxxx

xxxx

(3.3.1)

which of current conservation type ( 0 J ). This, however, implies the

existence of an algebraic consistency equation of Maxwell’s type ( JF )

relating the open-string variables ( L ) to the closed-string variables ( J ) in the

form(Maxwell 1959 p.69):

JLx (3.3.2)

Moreover, the open-string variables ( L ) are invariant under an internal SU(2)

group of transformations (equivalent to gauge-invariance of the second kind of

the electromagnetic field ( F )

xaxaxxaxax 22211211 ' ,' (3.3.3)

where 112212211 aaaa , i.e., LL ' ; and the representation of the closed

string variables ( J ) by 3x3 determinants via Eq.(3.2.9) implies their invariance

under an internal SU(3) group of transformations. Eq.(3.3.2) is the main result of

this sub-section. But this cannot be the end of the compatibility relations because

we have to deal with points 4M not on the light cone to which we now turn.

84


4. “EXTENDED” RELATIVITY ON THE TORUS AND

CONSTRAINTS ON DIMENSIONS

Since points not on the light cone arise from the quadratic nature of Eq(2.1.1c)

characterizing the torus, i.e.,

0)()(2 2222222222224 zyxtczyxtcss , (4.1)

we may characterize the torus as a net of three quadrics (Semple and Kneebone

1952, p. 339 ): namely, the quadric defined by the vanishing of the coefficient of 2s i.e.,

xxzyxtc 222220 (4.2a)

and the pair of quadrics defined by the remaining two equal roots of 4s

xxgzyxtcs )( 222222

, (4.2b)

where

1000

0100

0010

0001

1000

0100

0010

0001

g . (4.2c)

We recognize (4.2a) as a mutation of the light cone and Eq.(4.2b) as the standard

algebraic form of “extended” relativity principle. Moreover, if we use Dirac

matrices to characterize the metric tensors in Eq.(4.2c) as follows,

dxdxgdxdxdxdx

dxdxdxdxdxdxg

]2)3

(2)2

(2)1

(2)0

[(

2)3'(2)

2'(2)

1'(2)

0'('''

such that the transformation of the

g into 'g obeys the relation,

'

21''''

21' gg

(4.3a)

then the upper (+) sign in this relation is preserved by the transformation of the

vector (

) into an axial-vector (5

'

), giving the Lie-admissible

algebraic relation

85


15

UU

i.e., ,05

UU , (4.3b)

where )5

(2

1 IU , ( I25 ) and

32105 ..

For the geometric principle of duality to be applicable we must replace the

inhomogeneous algebraic relation, 02 sxxg

by a homogeneous one,

0244 sgxxg

so that the compatibility requirement implies the existence

of equivalent representations of (4.2b) in 5-dimensional space of 5-vectors,

)4,3,2,1,0( axa with sx 4 , or 10-dimensional space of the (5/2)(5-1)=10

independent components of the antisymmetric tensor abX which is the SUSY

partner of ax , or (Salam and Strathdee, 1982) 11-dimnsional Kaluza-Klein

supergravity space of 11 independent components of the symmetric metric tensor

(Semple and Kneebone, 1952. p. 388),

),,,,,.,,,,( 1231230302010011223344 ggggggggggg

As is well known, the constraints on the algebraic dimensions of a second-

quantized fermion field is determined by Fermi-Dirac statistics through the

representation of the anticommutation algebra of the single fermion creation and

annihilation operators, i

c and ic ,

,0'

,'

,

,'''',

jcic

jcic

ijic

jc

jc

ic

jcic

(4.4)

for i, j = 1, …, N, ),( . These operators are generators of an SO(2N) group

and can be represented by 2Nx2

N-matrices defined by Thouless (1961). The main

feature of the matrices is that the matrices decompose into N+1 blocks or

submatrices of dimension

i

N

i

N with N

N

i i

N2

1

(4.5)

where i =0,1,…,N. Consequently, the configuration is defined by the thK order

Pascal triangle coefficients of Nx )1( where N varies from 1 to K. The Pascal

triangle of dimensions is shown in Fig. 8 for N=1,2,…,8.

86


However, “fractal” dimension (Mandebrot 1983) can also be used to express

the compatibility relations for the points x not on the light cone as follows. We

notice that the 10-dimensional space of pair ( x , exex X ) has

compactification 64 BM in the form:

22

3

2

2

2

1

2

0 sxxxx ;

))(())(())(( 213013203210 xxxxxxxxxxxx (4.6)

if )1,1,1,1(),,,( 3210 eeee so that xx X , as in Eq.(3.5) above. Thus the

second equation in Eq.(4.6) can be the split into two cross-ratios:

2

1230

32102

2130

3120 1))((

))((,

))((

))((K

xxxx

xxxxK

xxxx

xxxx

(4.7)

where K is a “dimensionless separation parameter”. Each of the two cross-ratios

is invariant under the group of “fractal” (bilinear fractional) transformations

(Mandebrot 1983, p. 178):

)1( ),/()(' bcaddcxbaxxx (4.8)

Also the first cross-ratio in Eq.(4.7) has the rationalized form:

)()())(1(-

))(())((0

2

123

2

03212

31

2

02

2

212

23

2

01

2

21

2130

2

3120

XXXXKXXK

xxxxKxxxx

(4.9)

12 ,0 0 N ,

22 ,1 1 N

42 ,2 2 N

82 ,3 3 N

162 ,3 4 N

322 ,5 5 N

642 ,6 6 N

1282 ,7 7 N

2562 ,8 8 N 1 8 28 56 70 56 28 8 1 1 7 21 35 35 21 7 1

1 6 15 20 15 6 1 1 5 10 10 5 1

1 4 6 4 1 1 3 3 1

1 2 11 1

1

The Pascal Triangle for the 2Nx2

N matrix representation of SO(2N)

Fig.8

87


which is an SO(4,2) invariant form replacing the SO(3,3) form in Eq.(3.7) above.

It implies a compactification of the 10-dimensional space of the 10-vector

)X,( x to 64 BM characterized by the SO(1,3)xSO(4,2) of Poincare and

“fractal” symmetry transformations. Moreover, by representing points not on the

light cone in the parametric form, involving an “angular distance” ( ),

)sinh,sinh,sinh,(cosh),,,( 3210 isxxxx (4.10)

the cross-ratio takes the form

Kei

Kei

i

xxxx

xxxx i

log .,.

,)1)(sinh(cosh

)1)(sinh(cosh

))((

))(( 2)4/(2

2130

3120

(4.11)

where 4/ i is real. This takes the standard form of Mandelbrot’s relation

for fractal dimension (Mandebrot 1983, p. 37)

)/1log(/)log( rND (4.12)

on setting )log()/1( ND and rK /1 in Eq.(4.11). In other words,

compactification involving points in 4M not on the light cone demand the use of

non-integral (“fractal”) dimensions for consistency.

5. TEST OF THEORY THROUGH OBSERVATION OF

LIGHT CAUSTICS.

We turn finally to one of the problems with string theories, namely the fact that

their predictions in particle physics have not been testable (Kane, 2010). To

overcome this difficult in the present paper, we now proceed to outline how our

foundation of the principle of relativity on a torus in velocity space can be used to

predict the observation of light “caustics” in fiber optics. We begin by drawing an

analogy (in Fig.10) between the three different modes/geometries of optical fibers

highlighted in Nobel Prize (2009) and deep-inelastic electron-proton fusion of a

spherical p wavepacket into the hole of e torus as in the model of the so-

called Rutherford-Santilli(1978) neutron. The analogy enables us to associate the

two singularities in special relativity theory from the Lorentz transformation at

cv and from Einstein’s theorem on addition of velocities at 2cvV with

the rectilinear and wavepacket modes respectively of the three optical fiber

geometries produced by different geometries indicated in Fig 9.

88


By using the procedure adopted by Animalu and Ekuma (2008) for defining

Bjorken variable for scaling of structure functions for deep-inelastic e-p

scattering, we characterize the third mode in Fig.10 by the self-corresponding

points of a non-unitary transformation )'V,'v()V,v(

of a rectangular

hyperboloid into a torus:

Fig.9: Analogy between (Right) Deep-inelastic e-p fusion into

Rutherford-Santilli neutron and (Left) three different types of optical

fibers and corresponding modes, namely single mode (top), gradient

index multimode (middle) and step-index multimode (bottom) . The

propagation of a few rays is also indicated in red, as well as the

distribution geometry of the refractive index to the left. [From Two

revolutionary Optical Technologies compiled by the Class for

Physics of the Royal Swedish Academy of Sciences as scientific

background for the 2009 Nobel Prize in Physics].

89


)'V-'v(2

1V.v 22

, (5.1a)

i.e., 0V.v2v-V 22

(5.1b)

This is defined in terms of the parametric equations for a torus as follows:

.v.V2VvsinV2Vv0 i.e.,

;sinVu2uVvv v

;cosV v,sin)sinVu( v,cos)sinVu(

2222

222

3

2

2

2

1

321

ui

i

iiv

(5.2).

By rewriting Eq.(5.1a) in the matrix form

''')'V'v(v.V2 1

22

1 qqqq

. (5.3a)

we see that it involves a transformation of the underlying metric

, 10

01 '

01

1011

(5.3b)

such that under two (unitary and non-unitary) transformation matrices:

1 1 1 1

2 2 2 2-1 †

1 1 1 1

2 2 2 2

U ; U U

; 20 1

; 1 0

T T I

.(5.4a)

,0' 11 UTTU (5.4b)

has a Lie-admissible algebraic structure.

Moreover, if we re-express Eq.(5.1) in the form

.v.V2

2v2V

dv

dV

, (5.5)

we observe that it is separable (when V and v depend on a common local time

variable) into two equations for acceleration:

vV2k v

Eqn) s(Riccati' )vVk(V 22

.

(5.6)

But Eq. (5.5) can be integrated exactly to find

90


,v3

V

3

v3

0

22 V i.e., 3

0

32 vv3 VV (5.7a)

which may be expressed, on putting /v tr , as a cubic equation

23

0

3

3

0

32

)(3,)(),( where,0 i.e.,

,)()(3

VttVYXrYrX

tVrVtr

.. (5.8)

This states that one would see in (X,Y)-space, the envelope of the normals to a

parabola representing a “cusp” catastrophe (Poston and Stewart 1978) and shown

in Fig. 10a;

For an ellipse, the corresponding envelope of the normals (Fig.10b). can be seen

as “light caustics” generated by passing a laser beam through a grated louver

glass: this is one of the patterns that is a recurrent motif in the prehistoric African

(Igbo-Ukwu) bronze artefacts (Animalu 1990, Acholonu 2009).

6. DISCUSSION AND CONCLUSION

We have shown in this paper that, as envisaged by Green(1986), the space-time

geometry of the torus offers a natural geometry for formulating a Lorentz-

invariant string theory without divergences and we have seen that many of the

rich structure of string theory emerge from the conformal mapping of the torus

onto a two-sheeted (Riemann) plane available in conventional theory of Riemann

surfaces. The implication is that the universe may be a torus, as envisaged in

Fig.10: Envelope of the normals to (a) parabola , (b) ellipse.

91


Eq.(2.1.0), rather than a “cosmic egg” of the big bang theory. We have shown

how predictions can be made with our string theory in fiber optics and indicated

an analogy with deep-inelastic e-p scattering which we shall explore further in a

subsequent paper. .However, because current string theories including the one

described in this paper have non-unitary structure, which we have indicated in this

paper, it is necessary, as pointed out by Santilli(2002) ,to use the appropriate

mathematical structures of the Lie-admissible theory in order to avoid

catastrophic inconsistency.

ACKNOWLEDGEMENT

One of the authors, Late Charles N. Animalu(1966-2004), began this research

with the appearance in 1985 of M.B. Green’s popular Scientific American article

on Superstring Theory while he was a BS/MS student in the Department of

Electrical Engineering and Computer Science, Massachusetts Institute of

Technology, Cambridge, MA (U.S.A.). The work was suspended in 1991 to give

way for his Ph.D. research project in the Department of Physics and Astronomy,

University of Nigeria, on high-Tc superconductivity which was supported by an

award of Senate Research Grant No. 648 of the University of Nigeria, Nsukka.

After earning Ph.D.in 1992, the work was resumed in collaboration with his

father, Alex Animalu, who subsequently finalized it by 2010 in the present form

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space-time geometry of a torus for a lorentz- and conformal

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